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Lesson 1 How Do You Know That? Develop Understanding

Jump Start

Notice and Wonder

The following students have observed that the sum of two odd numbers seems to always be an even number, but their peers aren’t convinced that this is always the case. Each student has given a different explanation as to why they think this claim is correct.

Write at least two things you notice and one thing you wonder about the following explanations:

Student 1:

I heard one of my teachers say that the sum of two odd numbers is always even, and she knows everything about math.

Student 2:

I tried a lot of examples, and I found that the sum of two odd numbers is always even. For example, and .

Student 3:

I know that an odd number of objects can be paired together in groups of 2, with one object left over. If I combine two such groups together, the extra ones will also pair together, so the sum of two odd numbers will always be even. I drew the following diagram to show what I was thinking:

five pairs of squares and one single square in a group circled in black, three pairs of squares and one single square circled in black. The single squares are circled in green together.

Student 4:

Since an odd number is one more than an even number, I can represent odd numbers algebraically as or .

If I add two odd numbers together I get:

The last expression, , is even since it is times something. So, the sum of two odd numbers is always even.

Learning Focus

Examine ways of knowing that the sum of the angles in a triangle is .

How do I know something is true? Are there different ways that I know or accept things to be true?

When I notice a pattern in examples or through experimentation, how do I convince myself that my conjecture is always true?

Regardless of the shape or size of a triangle, is there a characteristic that is the same for all triangles in addition to being a three-sided polygon?

Open Up the Math: Launch, Explore, Discuss

You may know that the sum of the interior angles of any triangle is . (If you didn’t know, you do now!) But an important question to ask yourself is, “How do I know that?”

We know a lot of things because we accept it on authority—we believe what other people tell us; things such as the distance from the earth to the sun is or that the population of the United States is growing about each year. Other things are just defined to be so, such as the fact that there are in a mile. Some things we accept as true based on experience or repeated experiments, such as the sun always rises in the east, or “I get grounded every time I stay out after midnight.” In mathematics we have more formal ways of deciding if something is true.

Experiment #1

Cut out several triangles of different sizes and shapes. For each triangle, tear off the three corners (angles) and arrange the vertices so they meet at a single point, with the edges of the angles (rays) touching each other like pieces of a puzzle.

1.

What does this experiment reveal about the sum of the interior angles of the triangles you cut out, and how does it do so?

2.

Since you and your classmates have performed this experiment with several different triangles, does it guarantee that we will observe this same result for all triangles? Why or why not?

Experiment #2

Perhaps a different experiment will be more convincing. Cut out another triangle and trace it onto a piece of paper. It will be helpful to color-code each vertex angle of the original triangle with a different color. As new images of the triangle are produced during this experiment, color-code the corresponding angles with the same colors.

  • Locate the midpoints of each side of your cut out triangle by folding the vertices that form the endpoints of each side onto each other.

  • Rotate your triangle about the midpoint of one of its sides. Trace the new triangle onto your paper and color-code the angles of this image triangle so that corresponding image/pre-image pairs of angles are the same color.

  • Now rotate the new “image” triangle about the midpoint of one of the other two sides. Trace the new triangle onto your paper and color-code the angles of this new image triangle so that corresponding image/pre-image pairs of angles are the same color.

3.

What does this experiment reveal about the sum of the interior angles of the triangles you cut out, and how does it do so?

4.

Do you think you can rotate all triangles in the same way about the midpoints of its sides, and get the same results? Why or why not?

Ready for More?

Examining the Diagram

Experiment #2 produced a sequence of triangles, as illustrated in the diagrams.

Triangle ABC with Angle A with a green arc, angle B with a blue arc, and angle C with a red arc.
Triangle ABC with Angle A with a green arc, angle B with a blue arc, and angle C with a red arc. Triangle A'BC with Angle A' with green arc, angle B with red arc and angle C with blue arc. Triangle ABC and Triangle A'BC share side BC.
Triangle ABC with Angle A with a green arc, angle B with a blue arc, and angle C with a red arc. Triangle A'BC with Angle A' with green arc, angle B with red arc and angle C with blue arc. Triangle ABC and Triangle A'BC share side BC. Triangle A'CC" with Angle A' with blue arc, angle C with green arc, and angle C" with red arc. Triangle A'CC" and Triangle A'BC share side A'C.

Here are some things we might ask about this diagram:

  • Will the second figure in the sequence always be a parallelogram? Why or why not?

  • Will the last figure in the sequence always be a trapezoid? Why or why not?

Takeaways

Today we learned about four “Ways of Knowing”:

These ways of knowing showed up in our work today when we conjectured:

Which I now accept as true because: (responses will vary)

I noticed:

and I thought about

which led me to believe

So, my way of knowing this conjecture is true is

Types of Reasoning in Geometry:

Inductive reasoning: This type of reasoning consists of making conjectures based on experimentation across several examples.

Deductive reasoning: This type of reasoning consists of using:

  • properties,

  • definitions,

  • postulates (),

  • or theorems ().

Vocabulary

Lesson Summary

In this lesson, we explored different ways of knowing if something is true, such as basing our knowledge on being told by someone in authority, versus basing our knowledge on experimentation or reasoning with a diagram. We examined these ways of knowing in the context of justifying how we know that the sum of the angles in a triangle is .

Retrieval

1.

Use the diagram to determine which of the symbolic statements are correct and which are incorrect. Then explain why.

Triangle FEC with line segment EG with Point G on FC. Angle ECG is labeled with one arc, angle GEC with three arcs, angle F with two arcs, and angle EGC is a right angle. Triangle ACB with line segment CD with Point D on AB. Angle A is labeled with one arc, angle ACD with three arcs, angle F with two arc, and angle CDA is a right angle.

a.

b.

c.

d.

e.

f.

An angle that forms a straight line measures and can also be referred to as a straight angle. If two angles share a vertex and make a straight angle they are called a linear pair. This is the case because they make a straight line.

Use this information about a linear pair to find the measure of the missing angle.

2.

A line divided into two angles, one angle labeled 30 degrees. 30°

3.

A line divided into two angles, one angle labeled 140 degrees. 140°

4.

Two lines intersecting, one of the angles of intersection labeled 163 degrees. 163°